The Order Classes of 2-Generator -Groups

Algorithm 1

Theorem 7 in GAP’s algorithm.

p:=◯;;n:=◯;;order:=p    n;; # Input the values of p and n, where the order of G is p    n

G:=AllSmallGroups(Size,order);;

D:=NumberSmallGroups(order);;

for k in [1..D] do;

    f:=G[k];;m:=Size(MinimalGeneratingSet(f));;WW:=[ ];;

     if NilpotencyClassOfGroup(f)=2 and m=2 then;

      Add(WW,[1,1]);

      Print(k,") G=",StructureDescription(f), " |G|=",Size(f)," p=",p,"n=",n," N.class

",NilpotencyClassOfGroup(f));

       gg:=MinimalGeneratingSet(f);;

       e:=Identity(f);;

       a:=gg[1];;b:=gg[2];;

       i:=Log(Order(a),p);;j:=Log(Order(b),p);;w:=Maximum(i,j);;

        if i+j=n then;

         m1:=1; m2:=p    n/p    w-1;Add(WW,[p,m1*p    2-1]);

         if w>=2 then;

          Add(WW,[p    2,m2*p    2]);

         fi;

        else;

         m1:=p;m2:=p    (n-w)-p;Add(WW,[p,m1*p    2-1]);

         if w>=2 then;

          Add(WW,[p    2,m2*p    2]);

         fi;

        fi;

        Yw:=p    (n-1)*(p-1);;

        for 1 in [0..(w-3)] do;

         Add(WW,[p    (w-l),Yw/(p    1)]);

        od;

Print("∖n no. of gen.=",m,", o(a)=",p    i,", o(b)=",p    j,", w=",w,"∖n OC=",WW,"∖n∖n");

fi;od;time;